HMMT 十一月 2008 · 冲刺赛 · 第 17 题
HMMT November 2008 — Guts Round — Problem 17
题目详情
- [ 10 ] Determine the last two digits of 17 , written in base 10. 6
解析
- [ 10 ] Determine the last two digits of 17 , written in base 10. 17 Answer: 77 We are asked to find the remainder when 17 is divided by 100. Write the power as 17 (7 + 10) and expand with the binomial theorem: 17 17 16 (7 + 10) = 7 + 17 · 7 · 10 + . . . We can ignore terms with more than one factor of 10 because these terms are divisible by 100, so 4 adding them does not change the last two digits. Now, 7 = 2401, which has remainder 1 mod 100, so 17 16 7 has last two digits 07 and 7 · 10 has last two digits 70. We add these together. 6